On the optimal layout of planar graphs with fixed boundary
SIAM Journal on Computing
Parametrization and smooth approximation of surface triangulations
Computer Aided Geometric Design
Level set diagrams of polyhedral objects
Proceedings of the fifth ACM symposium on Solid modeling and applications
How to morph tilings injectively
Journal of Computational and Applied Mathematics
Cut-and-paste editing of multiresolution surfaces
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Metamorphosis of Arbitrary Triangular Meshes
IEEE Computer Graphics and Applications
One-to-one piecewise linear mappings over triangulations
Mathematics of Computation
Computer Aided Geometric Design
Tutte's barycenter method applied to isotopies
Computational Geometry: Theory and Applications - Special issue on the thirteenth canadian conference on computational geometry - CCCG'01
Matchmaker: constructing constrained texture maps
ACM SIGGRAPH 2003 Papers
Global conformal surface parameterization
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Hodge decompositions and computational electromagnetics
Hodge decompositions and computational electromagnetics
Discrete exterior calculus
Free-Boundary Linear Parameterization of 3D Meshes in the Presence of Constraints
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
Planar parameterization for closed 2-manifold genus-1 meshes
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Journal of Graph Theory
Discrete differential forms for computational modeling
ACM SIGGRAPH 2006 Courses
Geometric modeling based on triangle meshes
ACM SIGGRAPH 2006 Courses
Meshing genus-1 point clouds using discrete one-forms
Computers and Graphics
Surface parameterization via aligning optimal local flattening
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Design of tangent vector fields
ACM SIGGRAPH 2007 papers
Discrete laplace operators: no free lunch
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Mesh parameterization methods and their applications
Foundations and Trends® in Computer Graphics and Vision
Cycle bases of graphs and sampled manifolds
Computer Aided Geometric Design
Discrete quadratic curvature energies
Computer Aided Geometric Design
User-controllable polycube map for manifold spline construction
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Mesh parameterization: theory and practice
ACM SIGGRAPH ASIA 2008 courses
Generalized Koebe's method for conformal mapping multiply connected domains
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
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We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "spring-embedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.